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Convex Geodesic Bicombings and Hyperbolicity

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Convex Geodesic Bicombings and Hyperbolicity Convex geodesic bicombings and hyperbolicity Dominic Descombes & Urs Lang April 19, 2014 Abstract A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. We prove existence and uniqueness results for geodesic bicombings satisfying different convexity conditions. In combination with recent work by the second author on injective hulls, this shows that every word hyperbolic group acts geometrically on a proper, finite dimensional space X with a unique (hence equivariant) convex geodesic bicombing of the strongest type. Furthermore, the Gromov boundary of X is a Z-set in the closure of X, and the latter is a metrizable absolute retract, in analogy with the Bestvina–Mess theorem on the Rips complex. 1 Introduction By a geodesic bicombing σ on a metric space (X, d) we mean a map σ : X × X × [0, 1] → X that selects for every pair (x,y) ∈ X×X a constant speed geodesic σxy := σ(x,y, ·) ′ ′ ′ from x to y (that is, d(σxy(t),σxy(t )) = |t − t |d(x,y) for all t,t ∈ [0, 1], and σxy(0) = x, σxy(1) = y). We are interested in geodesic bicombings satisfying one of the following two conditions, each of which implies that σ is continuous and, hence, X is contractible. We call σ convex if the function t 7→ d(σxy(t),σx′y′ (t)) is convex on [0, 1] (1.1) for all x,y,x′,y′ ∈ X, and we say that σ is conical if arXiv:1404.5051v1 [math.MG] 20 Apr 2014 ′ ′ d(σxy(t),σx′y′ (t)) ≤ (1 − t) d(x,x )+ t d(y,y ) for all t ∈ [0, 1] (1.2) and x,y,x′,y′ ∈ X. Obviously every convex geodesic bicombing is conical, but the reverse implication does not hold in general (see Example 2.2). For this, in order to pass condition (1.2) to subsegments, one would need to know in addition that σ is consistent in the sense that σpq(λ)= σxy((1 − λ)s + λt) (1.3) 1 2 D. Descombes & U. Lang whenever x,y ∈ X, 0 ≤ s ≤ t ≤ 1, p := σxy(s), q := σxy(t), and λ ∈ [0, 1]. Thus, every conical and consistent geodesic bicombing is convex. Furthermore, we will say that σ is reversible if σxy(t)= σyx(1 − t) for all t ∈ [0, 1] (1.4) and x,y ∈ X. This property is not imposed here but will be obtained at no extra cost later. Basic examples of convex geodesic bicombings are given by the linear geodesics σxy(t) := (1 − t)x + ty in a linearly convex subset X of a normed space or by the unique geodesics σxy : [0, 1] → X in a CAT(0) space [2, 4] or a Busemann space [1, 20] (where t 7→ d(σ(t),τ(t)) is convex for every pair of geodesics σ, τ : [0, 1] → X.) An additional source of conical geodesic bicombings is the fact that this weaker notion behaves well under 1-Lipschitz retractions (see Lemma 2.1). Consistent and reversible geodesic bicombings have also been employed in [13, 10]. In view of the primary examples, the existence of a convex or conical geodesic bicombing on a metric space may be regarded as a weak (but non-coarse) global nonpositive curvature condition. One thus arrives at the following hierarchy of properties (A) ⇒ (B) ⇒ (C) ⇒ (D) ⇒ (E) for a geodesic metric space X: (A) X is a CAT(0) space; (B) X is a Busemann space; (C) X admits a convex and consistent geodesic bicombing; (D) X admits a convex geodesic bicombing; (E) X admits a conical geodesic bicombing. Clearly, if X is uniquely geodesic, then (E) ⇒ (B). When X is a normed real vector space, (A) holds if and only if the norm is induced by an inner product ([4, Proposition II.1.14]), (B) holds if and only if the norm is strictly convex ([20, Proposition 8.1.6]), and (C) is always satisfied. In the general case, (C) is stable under limit operations, whereas (B) is not (compare Sect. 10 in [17]). It is unclear whether (E) ⇒ (D) and (D) ⇒ (C) without further conditions. One of the purposes of this paper is to establish these implications under suitable assumptions, and to address questions of uniqueness. Our first result refers to (E) and (D). 1.1 Theorem. Let X be a proper metric space with a conical geodesic bicombing. Then X also admits a convex geodesic bicombing. The idea of the proof is to resort to a discretized convexity condition and then to gradually decrease the parameter of discreteness by the “cat’s cradle” construction from [1]. The passage from (D) to (C) appears to be more subtle. We first observe that the linear segments t 7→ (1 − t)x + ty in an arbitrary normed Convex geodesic bicombings and hyperbolicity 3 space X may be characterized as the curves γ : [0, 1] → X with the property that t 7→ d(z, γ(t)) is convex on [0, 1] for every z ∈ X (see Theorem 3.3). We therefore term curves with this property straight (in a metric space X). Since the geodesics of a convex bicombing are necessarily straight, the above observation shows that normed spaces have no such bicombing other than the linear one. By contrast, there are instances of non-linear straight geodesics in compact convex subspaces of normed spaces as well as multiple convex geodesic bicombings in compact metric spaces (see Examples 3.4 and 3.5). Nevertheless, we prove a strong uniqueness property of straight curves (Proposition 4.3) in spaces satisfying a certain finite dimensionality assumption, introduced by Dress in [7] and explained further below. This gives the following result regarding items (D) and (C). 1.2 Theorem. Let X be a metric space of finite combinatorial dimension in the sense of Dress, or with the property that every bounded subset has finite com- binatorial dimension. Suppose that X possesses a convex geodesic bicombing σ. Then σ is consistent, reversible, and unique, that is, σ is the only convex geodesic bicombing on X. Our interest in Theorems 1.1 and 1.2 comes from the fact that property (E) holds in particular for all absolute 1-Lipschitz retracts X. (To see this, embed X isometrically into ℓ∞(X) and retract the linear geodesics to X; compare again Lemma 2.1). These correspond to the injective objects in the category of metric spaces and 1-Lipschitz maps: X is injective if for every isometric inclusion A ⊂ B of metric spaces and every 1-Lipschitz map f : A → X there exists a 1-Lipschitz extension f¯: B → X of f. Basic examples include the real line, all ℓ∞ spaces, and all metric (R-)trees. Furthermore, by a 50-year-old result of Isbell [14], every metric space X possesses an essentially unique injective hull E(X). This yields a compact metric space if X is compact and a finite polyhedral complex with ℓ∞ metrics on the cells in case X is finite. Isbell’s explicit construction was redis- covered and further explored by Dress [7], who gave it the name tight span. The combinatorial dimension dimcomb(X) of a general metric space X is the supremum of the dimensions of the polyhedral complexes E(Y ) for all finite subspaces Y of X. In case X is already injective, every such E(Y ) embeds isometrically into X, so dimcomb(X) is bounded by the supremum of the topological dimensions of com- pact subsets of X. Dress also gave a characterization of the condition dimcomb ≤ n in terms of a 2(n + 1)-point inequality. We restate his result in Theorem 4.1 and provide a streamlined proof in an appendix to this paper. In [18], the second author proved that if Γ is a Gromov hyperbolic group, endowed with the word metric with respect to some finite generating set, then the injective hull E(Γ) is a proper, finite-dimensional polyhedral complex with finitely many isometry types of cells, isometric to polytopes in ℓ∞ spaces, and Γ acts properly and cocompactly on E(Γ) by cellular isometries. Furthermore, after barycentric subdivision, the resulting simplicial Γ-complex is a model for the classifying space EΓ for proper actions. Since X = E(Γ) satisfies property (E), 4 D. Descombes & U. Lang Theorems 1.1 and 1.2 together now show that E(Γ) possesses a convex geodesic bicombing that is consistent, reversible, and unique, hence equivariant with re- spect to the full isometry group of E(Γ). (The existence of an equivariant conical geodesic bicombing on E(Γ) was known before, see Proposition 3.8 in [18], but the proof of that fact did not give any indication on the consistency and uniqueness of the bicombing.) In particular, the following holds. 1.3 Theorem. Every word hyperbolic group Γ acts properly and cocompactly by isometries on a proper, finite-dimensional metric space X with a convex geodesic bicombing that is furthermore consistent, reversible, and unique, hence equivariant with respect to the isometry group of X. In a last part of the paper, we discuss the asymptotic geometry of complete metric spaces X with a convex and consistent geodesic bicombing σ. We define the geometric boundary ∂σX in terms of geodesic rays consistent with σ, and we equip Xσ = X ∪∂σX with a natural metrizable topology akin to the cone topology in the case of CAT(0) spaces. In view of Theorem 1.3, this unifies and generalizes the respective constructions for CAT(0) or Busemann spaces and for hyperbolic groups.
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